Base classes for reflexive modules¶

class sage.tensor.modules.reflexive_module.ReflexiveModule_abstract[source]¶

Bases: Parent

Abstract base class for reflexive modules.

An \(R\)-module \(M\) is reflexive if the natural map from \(M\) to its double dual \(M^{**}\) is an isomorphism.

In the category of \(R\)-modules, the dual module \(M^*\) is the \(R\)-module of linear functionals \(\phi:\ M \longrightarrow R\). However, we do not make the assumption that the dual module (obtained by dual()) is in the category Homsets.

We identify the double dual \(M^{**}\) with \(M\).

Tensor products of reflexive modules are reflexive. We identify all tensor products of \(k\) copies of \(M\) and \(l\) copies of \(M^*\) and denote it by \(T^{(k,l)}(M)\). The tensor_type() of such a tensor product is the pair \((k, l)\), and \(M\) is called its base_module().

There are three abstract subclasses:

ReflexiveModule_base is the base class for implementations of base modules \(M\).
ReflexiveModule_dual is the base class for implementations of duals \(M^*\).
ReflexiveModule_tensor is the base class for implementations of tensor modules \(T^{(k,l)}(M)\).

base_module()[source]¶

Return the module on which self is constructed.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3)
sage: M.base_module() is M
True
sage: M.dual().base_module() is M
True
sage: M.tensor_module(1, 2).base_module() is M
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3))
>>> M.base_module() is M
True
>>> M.dual().base_module() is M
True
>>> M.tensor_module(Integer(1), Integer(2)).base_module() is M
True

dual()[source]¶

Return the dual module.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3)
sage: M.dual()
Dual of the Rank-3 free module over the Integer Ring
sage: M.dual().dual()
Rank-3 free module over the Integer Ring
sage: M.tensor_module(1, 2)
Free module of type-(1,2) tensors on the Rank-3 free module over the Integer Ring
sage: M.tensor_module(1, 2).dual()
Free module of type-(2,1) tensors on the Rank-3 free module over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3))
>>> M.dual()
Dual of the Rank-3 free module over the Integer Ring
>>> M.dual().dual()
Rank-3 free module over the Integer Ring
>>> M.tensor_module(Integer(1), Integer(2))
Free module of type-(1,2) tensors on the Rank-3 free module over the Integer Ring
>>> M.tensor_module(Integer(1), Integer(2)).dual()
Free module of type-(2,1) tensors on the Rank-3 free module over the Integer Ring

tensor(*args, **kwds)[source]¶

Return the tensor product of self and others.

This method is invoked when TensorProductFunctor is applied to parents.

It just delegates to tensor_product().

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(QQ, 2); M
2-dimensional vector space over the Rational Field
sage: M20 = M.tensor_module(2, 0); M20
Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
sage: tensor([M20, M20])
Free module of type-(4,0) tensors on the 2-dimensional vector space over the Rational Field

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(QQ, Integer(2)); M
2-dimensional vector space over the Rational Field
>>> M20 = M.tensor_module(Integer(2), Integer(0)); M20
Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
>>> tensor([M20, M20])
Free module of type-(4,0) tensors on the 2-dimensional vector space over the Rational Field

tensor_power(n)[source]¶

Return the n-fold tensor product of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(QQ, 2)
sage: M.tensor_power(3)
Free module of type-(3,0) tensors on the 2-dimensional vector space over the Rational Field
sage: M.tensor_module(1,2).tensor_power(3)
Free module of type-(3,6) tensors on the 2-dimensional vector space over the Rational Field

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(QQ, Integer(2))
>>> M.tensor_power(Integer(3))
Free module of type-(3,0) tensors on the 2-dimensional vector space over the Rational Field
>>> M.tensor_module(Integer(1),Integer(2)).tensor_power(Integer(3))
Free module of type-(3,6) tensors on the 2-dimensional vector space over the Rational Field

tensor_product(*others)[source]¶

Return the tensor product of self and others.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(QQ, 2)
sage: M.tensor_product(M)
Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
sage: M.tensor_product(M.dual())
Free module of type-(1,1) tensors on the 2-dimensional vector space over the Rational Field
sage: M.dual().tensor_product(M, M.dual())
Free module of type-(1,2) tensors on the 2-dimensional vector space over the Rational Field
sage: M.tensor_product(M.tensor_module(1,2))
Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
sage: M.tensor_module(1,2).tensor_product(M)
Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
sage: M.tensor_module(1,1).tensor_product(M.tensor_module(1,2))
Free module of type-(2,3) tensors on the 2-dimensional vector space over the Rational Field

sage: Sym2M = M.tensor_module(2, 0, sym=range(2)); Sym2M
Free module of fully symmetric type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
sage: Sym01x23M = Sym2M.tensor_product(Sym2M); Sym01x23M
Free module of type-(4,0) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (0, 1), with symmetry on the index positions (2, 3)
sage: Sym01x23M._index_maps
((0, 1), (2, 3))

sage: N = M.tensor_module(3, 3, sym=[1, 2], antisym=[3, 4]); N
Free module of type-(3,3) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (1, 2),
 with antisymmetry on the index positions (3, 4)
sage: NxN = N.tensor_product(N); NxN
Free module of type-(6,6) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (1, 2), with symmetry on the index positions (4, 5),
 with antisymmetry on the index positions (6, 7), with antisymmetry on the index positions (9, 10)
sage: NxN._index_maps
((0, 1, 2, 6, 7, 8), (3, 4, 5, 9, 10, 11))

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(QQ, Integer(2))
>>> M.tensor_product(M)
Free module of type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
>>> M.tensor_product(M.dual())
Free module of type-(1,1) tensors on the 2-dimensional vector space over the Rational Field
>>> M.dual().tensor_product(M, M.dual())
Free module of type-(1,2) tensors on the 2-dimensional vector space over the Rational Field
>>> M.tensor_product(M.tensor_module(Integer(1),Integer(2)))
Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
>>> M.tensor_module(Integer(1),Integer(2)).tensor_product(M)
Free module of type-(2,2) tensors on the 2-dimensional vector space over the Rational Field
>>> M.tensor_module(Integer(1),Integer(1)).tensor_product(M.tensor_module(Integer(1),Integer(2)))
Free module of type-(2,3) tensors on the 2-dimensional vector space over the Rational Field

>>> Sym2M = M.tensor_module(Integer(2), Integer(0), sym=range(Integer(2))); Sym2M
Free module of fully symmetric type-(2,0) tensors on the 2-dimensional vector space over the Rational Field
>>> Sym01x23M = Sym2M.tensor_product(Sym2M); Sym01x23M
Free module of type-(4,0) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (0, 1), with symmetry on the index positions (2, 3)
>>> Sym01x23M._index_maps
((0, 1), (2, 3))

>>> N = M.tensor_module(Integer(3), Integer(3), sym=[Integer(1), Integer(2)], antisym=[Integer(3), Integer(4)]); N
Free module of type-(3,3) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (1, 2),
 with antisymmetry on the index positions (3, 4)
>>> NxN = N.tensor_product(N); NxN
Free module of type-(6,6) tensors on the 2-dimensional vector space over the Rational Field,
 with symmetry on the index positions (1, 2), with symmetry on the index positions (4, 5),
 with antisymmetry on the index positions (6, 7), with antisymmetry on the index positions (9, 10)
>>> NxN._index_maps
((0, 1, 2, 6, 7, 8), (3, 4, 5, 9, 10, 11))

tensor_type()[source]¶

Return the tensor type of self.

OUTPUT:

pair \((k,l)\) such that self is the module tensor product \(T^{(k,l)}(M)\), where \(M\) is the base_module() of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3)
sage: T = M.tensor_module(1, 2)
sage: T.tensor_type()
(1, 2)

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3))
>>> T = M.tensor_module(Integer(1), Integer(2))
>>> T.tensor_type()
(1, 2)

class sage.tensor.modules.reflexive_module.ReflexiveModule_base[source]¶

Bases: ReflexiveModule_abstract

Abstract base class for reflexive modules that are base modules.

base_module()[source]¶

Return the free module on which self is constructed, namely self itself.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: M.base_module() is M
True

sage: M = Manifold(2, 'M')                                                  # needs sage.symbolic
sage: XM = M.vector_field_module()                                          # needs sage.symbolic
sage: XM.base_module() is XM                                                # needs sage.symbolic
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> M.base_module() is M
True

>>> M = Manifold(Integer(2), 'M')                                                  # needs sage.symbolic
>>> XM = M.vector_field_module()                                          # needs sage.symbolic
>>> XM.base_module() is XM                                                # needs sage.symbolic
True

dual()[source]¶

Return the dual module.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: M.dual()
Dual of the Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> M.dual()
Dual of the Rank-3 free module M over the Integer Ring

tensor_module(k, l, **kwds)[source]¶

Return the module of all tensors of type \((k, l)\) defined on self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3)
sage: M.tensor_module(1, 2)
Free module of type-(1,2) tensors on the Rank-3 free module over the Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3))
>>> M.tensor_module(Integer(1), Integer(2))
Free module of type-(1,2) tensors on the Rank-3 free module over the Integer Ring

tensor_type()[source]¶

Return the tensor type of self, the pair \((1, 0)\).

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3)
sage: M.tensor_type()
(1, 0)

sage: M = Manifold(2, 'M')                                                  # needs sage.symbolic
sage: XM = M.vector_field_module()                                          # needs sage.symbolic
sage: XM.tensor_type()                                                      # needs sage.symbolic
(1, 0)

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3))
>>> M.tensor_type()
(1, 0)

>>> M = Manifold(Integer(2), 'M')                                                  # needs sage.symbolic
>>> XM = M.vector_field_module()                                          # needs sage.symbolic
>>> XM.tensor_type()                                                      # needs sage.symbolic
(1, 0)

class sage.tensor.modules.reflexive_module.ReflexiveModule_dual[source]¶

Bases: ReflexiveModule_abstract

Abstract base class for reflexive modules that are the duals of base modules.

construction()[source]¶

Return the functorial construction of self.

This implementation just returns None, as no functorial construction is implemented.

tensor_type()[source]¶

Return the tensor type of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: M.dual().tensor_type()
(0, 1)

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> M.dual().tensor_type()
(0, 1)

class sage.tensor.modules.reflexive_module.ReflexiveModule_tensor[source]¶

Bases: ReflexiveModule_abstract

Abstract base class for reflexive modules that are tensor products of base modules.

tensor_factors()[source]¶

Return the tensor factors of this tensor module.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: T = M.tensor_module(2, 3)
sage: T.tensor_factors()
[Rank-3 free module M over the Integer Ring,
 Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring]

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> T = M.tensor_module(Integer(2), Integer(3))
>>> T.tensor_factors()
[Rank-3 free module M over the Integer Ring,
 Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring,
 Dual of the Rank-3 free module M over the Integer Ring]